Derivative functions

Exercise 1

Use differentiation to find all inflection points of the following functions.

a

$f (x) = 0,5 x^{3} + 6 x^{2} - 90$

b

$y (x) = 4 x^{2} - 0,5 x^{4}$

Exercise 2

Given two functions $f (x) = x^{2}$ and $g (x) = 0,25 x^{2} (x^{2} - 144)$ .

a

Here you see how your graphing calculator can show the graphs of the two functions. What settings do you need to use to get this picture?

b

Solve: $f (x) > g (x)$

c

Calculate the exact inflection points of the graph of $g$ .

d

In the graph of $g$ , the tangents at the two inflection points intersect on the $y$ -axis. Calculate the exact coordinates of this intersect.

Exercise 3

An enterpreneur has a monopoly on the manufacture of a certain product. His production costs (in hundreds of euros) are given by the formula $T C = 0,5 q^{3} - 4 q^{2} + 11 q + 4$ where $q$ is the amount produced in hundreds of kilograms.

a

The speed with which the costs are rising is initially decreasing, but subsequently increasing. At one point in the graph, the speed therefore changes from decreasing to increasing.
At what production volume do you see this changeover point? Round your answer to whole kilograms.

b

The amount of product the manufacturer offers to his customers influences the price, according to the formula: $p = 11 - q$ where $p$ is the price in hundreds of euros. You can assume that the manufacturer always sells his entire production volume. At what production volume does he make the highest profit? Use differentiation to prove your answer.

Exercise 4

This is the graph of the first derivative of a certain function.

a

At what values of $x$ does this function have extrema?

b

At what values of $x$ does the graph of the function have an inflection point?

c

Does the tangent at the inflection point have a positive or a negative slope?

Exercise 5

Given the function $f (x) = x^{4} + a x^{2}$ with a constant $a > 0$ .

a

Show that this function has a minimum for any value of $a$ .

b

Show that this function has no inflection points for any value of $a$ .

Exercises